M ar 1 99 9 Effective temperatures out of equilibrium ∗

We describe some interesting effects observed during the evolution of nonequilibrium systems, using domain growth and glassy systems as examples. We breafly discuss the analytical tools that have been recently used to study the dynamics of these systems. We mainly concentrate on one of the results obtained from this study, the violation of the fluctuation-dissipation theorem and we discuss, in particular, its relation to the definition and measurement of effective temperatures out of equilibrium. One of the major challenges in physics is to understand the behaviour of systems that are far from equilibrium. These systems are ubiquitous in nature. Some examples are phase separation, systems undergoing domain growth, all types of glasses, turbulent flows, systems driven by non-potential forces, etc. All these systems are “large” in the sense that they are composed of many, N → ∞, dynamic degrees of freedom. Apart from succeeding in predicting the time evolution of their macroscopic properties, one would like to know which, if any, of the thermodynamic notions apply to these nonequilibrium cases. Systems undergoing domain growth, or phase separation, provide the best known example of a nonequilibrium evolution. Take for instance a magnetic system with ferromagnetic interactions in contact with a thermal bath. If the bath temperature is very high the sample is in its paramagnetic phase and the magnetic moments, or spins, point in random directions. If one next cools down the bath, and hence the sample, through a transition temperature Tc, the system enters the low temperature phase and starts forming domains or islands of the two ordered phases, say up and down. For definiteness, let us fix the final temperature to be 0 < T < Tc. At any time tw after crossing the transition at the initial time, two types of dynamics appear: (i) fast fluctuations of some spins, due to thermal fluctuations, Based on talks given at “Trends in Theoretical Physics II”, November 30 December 4, 1998, Buenos Aires, Argentina and in the NATO Advanced Study Institute “Topological Defects and the Non-Equilibrium Dynamics of Symmetry Breaking Phase Transitions”, February 16 26, 1999, Les Houches, France. LPTENS/9910